1. USE OF BASIC STATISTICS IN INSURANCE PRICING AND RISK ASSESSMENT
March 2017

2. Agenda
Statistical concepts relating to insurance: (1) Risk data (2) Presentation of risk data (3) Statistical measurement (4) Deriving probability (5) Probability distributions (6) Regression and correlation Practical examples to illustrate the above concepts

3. (1) Risk data

4. (i) using published data
Risk data
Database
Use existing database, or
Create a database by:
(i) using published data
(ii) gathering data - direct observation
- interviews
- experiments
- questionnaires

5. Risk data
Compare:
Population
Stratified sampling
Random sampling

6. (2) Presentation of risk data

7. Presentation of risk data
TABLES
Simple tables
Compound tables
Relative figures
Frequency and severity
Effect of mix change

8. Presentation of risk data
Claims Analysis:
a comparison of frequency and severity
Cost (£) Frequency % Severity %
0<1, , % £3.00m %
<2, , % £4.25m %
<3, , % £3.10m %
<4, % £1.75m %
<5, % £1.45m %
Total , % £13.55m %

9. Presentation of risk data
Effect of mix change
It is important to look at the elements which make up a total, not just the total itself

10. Presentation of risk data
GRAPHS
Single line graphs
Multiple line graphs
X-Y graphs

11. Presentation of risk data
BAR CHARTS
Simple bar charts
Multiple bar charts
Component bar charts

12. Presentation of risk data
PIE CHARTS
Simple pie charts
Enhanced pie charts
PICTOGRAMS

13. FREQUENCY DISTRIBUTIONS
Presentation of risk data
FREQUENCY DISTRIBUTIONS
Unordered array
Ordered array
Frequency distribution
Group frequency distribution

14. Presentation of risk data
To prepare a grouped frequency distribution from raw data:
the highest and lowest values (the range)
decide upon the number of classes (usually 4, 5 or 6)
divide the entire range by the number of classes
decide how to display the data (continuous or discrete;
size of classes)
allocate the individual values to the classes

15. Presentation of risk data
Discrete Variables can only be whole numbers, fractions are impossible. Therefore they can be represented by class limits of 0-9; 10-19; and so on.
Continuous Variables can be any fraction of a number, represented by class limits 0<10, 10<20 and so on.

16. Presentation of risk data
Relative Frequency the percentage of recorded figures in each class as a measure of the total number of raw data items.
Cumulative frequency the addition of each frequency class to the total of its predecessors. Compare “less than” and “more than”
Histogram the area represents the actual values for each frequency distribution. Horizontal axis (continuous scale) records class limits, vertical axis records frequency values

17. Presentation of risk data
Relative and Cumulative Frequencies
Cost Frequency Relative Cumulative
Frequency Frequency
0< % 20 40
150< % 30 20
300< % 35 10
450< %
600< %
Total %

18. Presentation of risk data
Ogives are graphs of cumulative frequency distributions horizontal axis represents class limits vertical axis represents cumulative frequency
Frequency polygon
add a class interval at each end of the histogram
mark the mid points at the tops of each of the
distribution rectangles
join the mid points with a straight line
the area of a frequency polygon = area of the histogram
Frequency curve is a free-hand drawing of shape of the curve

19. (3) Statistical measurement

20. Statistical measurement
Location - where the data is placed in the whole span of possible data variables available
Dispersion - the extent to which individual items in a set of values differ from each other in magnitude
Skew - the clustering of the data around the mean, median and mode

21. Statistical measurement
LOCATION
Arithmetic Mean
Geometric Mean
Median
Mode

22. Statistical measurement
Arithmetic mean for list of values
Add up all the data items and dividing by the number of items
Arithmetic mean = x n

23. Statistical measurement
Arithmetic mean for frequency distribution
e.g. time it takes to settle a claim in ABC Insurance plc Time in days Number of claims
x f fx
,000
,365
,040
f = fx = 4,140
Arithmetic mean = fx = 4,140 = 23 days f

24. Statistical measurement
Arithmetic mean for grouped data
Travel claims costs
£ No
150<160 10
160<170 20
170<180 15
180<190 12
mid-pts
f x fx
150<
160<
170<
180<
_ 57 9,695
So x = f x = 9,695 = £170.09 f

25. Statistical measurement
Important Features of the Arithmetic mean
Most commonly used form of average
Involves all values in a distribution
Advantage: completeness
Disadvantage: easily distorted by extreme values.
To avoid this disadvantage, use the median or the mode
Used to work out the standard deviation
Can produce ‘impossible values’ in the case of discrete data
(example: average of 2.1 children in a family)
Not applicable where relative changes are being averaged. In
such cases the geometric mean must be used.

26. Statistical measurement
Geometric mean
Used when relative changes in one variable are being averaged

27. Statistical measurement
Median for list of values
(1) Arrange the data values in numerical order
(2) For odd-numbered array of values, take the middle number
(3) For even-numbered array of values, take the two middle values and calculate the arithmetic mean

28. Statistical measurement
Median for a frequency distribution
No. of clerks Frequency Cumulative Frequency 29
Median number of clerks is 29

29. Statistical measurement
Median for grouped data
Claims costs (£) Frequency Cumulative Frequency
100

30. Statistical measurement
Finding the median for grouped data
Find the class having the median value in it
Decide on the class boundaries
Find out how far in to the class you must go (consult
the cumulative frequency distribution)
Determine the class interval and multiply this by the
distance you wish to move into it
Add this to the lower boundary of the class
See formula

31. Statistical measurement
Mode for list of values
The most frequently occurring value
Mode for grouped data
Mode = mean – 3(mean – median)

32. Statistical measurement
DISPERSION
Important to know the distribution or variance of the data values around the location
Range
the difference between the highest and lowest values.
Quartile deviation
The difference between the top and bottom quarters of the values indicates the inter-quartile range. Divide by 2 to get quartile deviation.

33. Statistical measurement
Standard Deviation
the most satisfactory measure of distribution as it uses all the values
Coefficient of variation
used to compare the relative variability, or dispersion, of two or more sets of figures

34. Statistical measurement
Standard deviation
For a simple list: _
s= (x- x)2 n
For grouped data (adapted for midpoints)
s= f(x- x)2 f
s= fx f x 2
f f

35. Statistical measurement
SKEW
Distribution with many small values (example: claims under household policies)
Symmetrical
Distribution:
Distribution with many large values (example: claims aviation insurance)
The difference between mean and median can be used to measure Skew:
Skew = 3 (mean - median)
standard deviation

36. Statistical measurement
Normal distribution a frequency distribution which shows a symmetrical curve peaking at the centre.
Mean median and mode coincide
Positive Distribution the peak lies to the left of centre;
the mean is dragged to the right due to high outliners;
the mode is the peak of the distribution
Negative Distribution the peak lies to the right of centre;
the mean has been dragged to the left due to very low outliners;
Pearson’s Coefficient measures the degree of skew – I.e. how far
of Skewness the mean is from the mode;
the lower the value calculated, the nearer the data is to a normal distribution;
formula automatically gives the direction of skew
mean
mode

37. (4) Deriving probability

38. Deriving probability PROBABILITY
The contributions of the many will be used to pay for the losses of the few
To know how much each insured must contribute to the pool, we must know what the likely losses are going to be
Probability theory is a formal mechanism for measuring likelihood

39. Deriving probability Deriving probabilities
A priori probability is one that applies when all the possible outcomes of the event are known before the event occurs
Relative Frequency based upon empirical (historical) information of what has happened in the past
Subjective probability deal with occurrences that have not happened at all before or very infrequently

40. Deriving probability Probability rules
Alternative Events P(A or B) – “additions” rule
(1) Events mutually exclusive
(2) Events not mutually exclusive - need to remove the aspect of double counting (best visualised by Venn Diagrams)
Joint Events P(A and B) – “multiplication” rule
(1) Independent events
(2) Dependent events

41. Deriving probability Probability rules Alternative events:
Mutually exclusive
P(A or B) = P(A) + P(B)
Not Mutually exclusive
P(A or B) = P(A) + P(B) - P(A and B)
Joint Events:
Independent
P(A and B) = P(A) x P(B)
Dependent
P(A and B) = P(A) x P(B/A)

42. (5) Probability distributions

43. Probability distributions
Expected value
When tossing a coin 100 times you would expect to get 50 heads because ½ times 100 = 50
When rolling a die 60 times you would expect 10 sixes because 1/6 times 60 = 10
 Expected value E = P(x).x
In frequency distributions the expected values of the various outcomes have to be added to find the total: E = P(x).x

44. Probability distributions
Expected Frequency
Number Number Probability Expected number of
of thefts of shops distribution thefts per shop
x P(x) P(x).x 1,

45. Probability distributions
Law of large numbers
The actual number of events occurring will tend towards the expected number where there are a large number of similar situations

46. Probability distributions
Expected severity
Cost Frequency Probability Midpoint Expected
per theft (£) distribution cost loss per theft P(x) x P(x).x
0<
300<
600<
900<1,
1,200<1,

47. Probability distributions
Premium calculation:
Expected number of thefts
per shop: 0.45
Expected loss per theft: £315.00
Pure premium per shop: £315 x 0.45 = £141.75

48. Probability distributions
NORMAL DISTRIBUTION
Symmetrical bell shaped curve
Mean under the apex, coincides with mode and median
Tails never touch the horizontal
Specific areas around the mean can be measured

49. Probability distributions
The width of the bell depends upon the actual spread of the distribution of data measured by the Standard Deviation.
Data with the same mean but different Standard Deviations will produce different curves
Knowing the mean and standard distribution for different groups of data enable comparisons to be made between them and a normal distribution curve

50. Probability distributions
Percentages under the normal curve
34.13%
13.59%
2.15%
0.13%
-3
-2
-1 
+1
+2
+3
68.26%
95.44%
99.74%
34.13%
34.13%
13.59%
13.59%
2.15%
2.15%
0.13%
0.13%
-3
-2
-1 
+1
+2
+3
68.26%
95.44%
99.74%

51. (6) Regression and correlation

52. Regression and correlation
Regression indicates what kind of relationship there is between dependent and independent variables.
Correlation measures the strength of that relationship

53. Regression and correlation
y = a + bx
Where y is the dependent variable
a is the where the line crosses the vertical axis
b is the gradient of the line (regression coefficient)
x is the independent variable
a and b are constant

54. Regression and correlation
Regression formulae
Line of best fit
y = a + bx
_ _
xy – nxy
Where b = _
x2 - nx2
and a = y - bx n

55. Regression and correlation
The equation y = a+bx is true for all straight lines. By calculating values of ‘a’ and ‘b’ from the data provided for ‘x’ and ‘y’, the line that minimises the total of the squared deviations of all points from it will be established. This is the line calculated according to the least squares method.
Regression coefficient, the value of ‘b’, illustrates the nature of the relationship
positive value for ‘b’ shows that as ‘x’ gets bigger so does ‘y’
negative value show that as ‘x’ gets bigger ‘y’ gets smaller
Can predict values for ‘y’ from any value of ‘x’ between the limits of ‘x’ provided in the data only

56. Regression and correlation
Correlation = a measure of the strength of the relationship between two variables
Coefficient of determination
r2 = explained variation
total variation
expressed as a percentage
Correlation coefficient
r = square root of r2
figures between +1 and -1

57. Regression and correlation
= how much of the variation in the predicted variable
is due to its relationship with the other variable

58. Regression and correlation
Correlation coefficient (r) or
-1  r  1
i.e. r is between -1 and 1

59. Regression and correlation
Coefficient of determination
Measures how much of the variation in the predicted variable is due to its relationship with the other variable
Expressed as a percentage
Low values suggest that the relationship is so low that it is not worth considering
Correlation coefficient
Assess on scale +1 through zero to -1
The closer ‘r’ is to 1, the closer the relationship between the variables but it does NOT mean causation
A score of +1 represents a perfect positive correlation - increases in ‘x’ result in increases in ‘y’
A score of -1 represents a perfect negative correlation - increases in ‘x’ result in decreases in ‘y’
A score of 0 means there is no correlation - changes in one variable have no effect on the other

60. Regression and correlation